Question: The measures of the interior angles of a particular triangle are in a 5:6:7 ratio. What is the measure, in degrees, of the smallest interior angle?
Answer: Choose $k$ so that the smallest angle measures $5k$ degrees.  Then the measures of the other two angles are $6k$ degrees and $7k$ degrees.  Since the measures of the angles in a triangle sum to 180 degrees, we have $5k+6k+7k=180\implies 18k=180\implies k=10$.  The smallest angle measures $5k=5(10)=\boxed{50}$ degrees.